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Deep circulation flows
Storyboard

There are several points where flows from the ocean surface to greater depths occur, inducing deep circulation. This circulation is subject to the Coriolis force, resulting in deviations and some flows towards the surface (upwelling) that are associated with surface currents.

The classic model for these currents is that of Stommel and Arons, which, although simple, explains the different depth flows observed.

[1] Ocean Circulation Theory, Joseph Pedlosky, Springer 1998 (7.3 Stommel-Arons Theory: Abyssal Flow on the Sphere)

>Model

ID:(1623, 0)


Thermohaline circulation over the planet
Note

If we observe the globe, the thermohaline circulation is generated near one of the poles (north or south) through water that, due to higher salinity and lower temperature, begins to sink. Its flow is directed towards the equator, creating an upwelling where water partially rises and flows towards the pole to replenish the descending water.

Representation of the North Atlantic in the Stommel and Arons model [1], [2]

[1] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanI. Stationary planetary flow patterns on a sphere. Deep Sea Research (1953), 6(2), 140-154.

[2] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanII. An idealized model of the circulation pattern and amplitude in oceanic basins. Deep Sea Research (1953), 6(3), 217-233.

ID:(12096, 0)


Box model
Quote

The Stommel and Arons model [1], [2] considers the ocean as a two-dimensional box with coordinates on the x and y axes. Specifically:

- Coordinates on the x-axis: $x_w$ (west) and $x_e$ (east).
- Coordinates on the y-axis: $y_s$ (south) and $y_n$ (north).

These coordinates are represented in the following graph:

Atlantic box model [1], [2].

[1] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanI. Stationary planetary flow patterns on a sphere. Deep Sea Research (1953), 6(2), 140-154.

[2] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanII. An idealized model of the circulation pattern and amplitude in oceanic basins. Deep Sea Research (1953), 6(3), 217-233.

ID:(12082, 0)


Characteristic times
Exercise

Each stage is associated with a characteristic time:

- Travel time with the main flow $\Delta t_y$
- Deflection time with the loss flow $\Delta t_x$
- Upwelling time $\Delta t_z$

ID:(13426, 0)


Velocities and Accelerations per Flow
Equation

Each characteristic time is associated with velocities and accelerations along the traveled path:

- With the main flow $v_y, a_y$.
- With the loss flow $v_x, a_x$.
- With the upwelling $v_z, a_z$.

Generally, the initial velocity ($v_y$) triggers, through the Coriolis force, the accelerations that lead to loss and upwelling.

ID:(13427, 0)


Lost Flow Geometry
Script

The loss flow is not uniform and is distributed along the latitude, so it is modeled based on its distance from the northernmost position. Thus, it is zero at northern latitudes and maximum at the southern edge of the rectangle where the circulation is modeled:

ID:(13428, 0)


Upwelling flow geometry
Variable

Since the loss flow is not uniform, neither is the upwelling. Within the same model, it is assumed that the upwelling is maximum at the eastern edge of the rectangle where the circulation is modeled. Similar to the loss, a linear relationship is assumed:

ID:(13429, 0)


Main flows of deep currents
Audio

There are four flows to consider in modeling deep flow:

The main flow $F_w$, which moves along the seafloor.
The loss flow $F_i$, which is the fraction deviated due to the Coriolis force.
The upwelling flow $U_x$, which corresponds to the fraction of loss flow reaching the surface.
The sinking flow $S_0, originating from surface currents, including losses that sink again.

ID:(13425, 0)


Underwater currents and Coriolis
Video

The so-called Coriolis force plays an essential role in the dynamics of water in the poles, influencing how water masses descend due to variations in temperature and salinity.



When analyzing the Atlantic Ocean, one can observe a movement of water from the pole towards the equator, which deflects towards the west. This phenomenon is caused by the lag relative to the planet\'s rotation, as it transitions from a zone of slower speed along the latitude to one of higher speed. This behavior can be modeled using the Coriolis equation for the x-direction, given by coriolis acceleration on the surface, in the x direction $m/s^2$, coriolis factor $rad/s$ and speed in meridian $m/s$:

$ a_{c,x} = f v_y $



In this equation, the Coriolis factor f is positive in the Northern Hemisphere and negative in the Southern Hemisphere, generating a tendency for the current to \'approach\' the American continent.

The geographical contour of the continent allows for movement in the x-direction (longitude), resulting in an acceleration in the y-direction (latitude), which can be calculated using coriolis acceleration on the surface, in the y direction $m/s^2$, coriolis factor $rad/s$ and parallel speed $m/s$:

$ a_{c,y} = - f v_x $



This calculation reveals that near the equator, displacements occur that take water away from the main current, moving it northward. If we examine the acceleration in the z-direction (depth) and take into account that \beta also changes sign with the hemisphere, the result is positive. In other words, an upwelling is observed, which can be estimated using coriolis acceleration in z direction $m/s^2$, coriolis Beta Factor $rad/s m$, parallel speed $m/s$ and planet radio $m$:

$ a_{cz} = R \beta v_x $

.

ID:(12122, 0)


Stommel-Arons depth flows
Unit

At the end, Stommel and Arons [1], [2] solve the model, indicating the main deep flows that exist throughout the globe:

[1] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanI. Stationary planetary flow patterns on a sphere. Deep Sea Research (1953), 6(2), 140-154.

[2] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanII. An idealized model of the circulation pattern and amplitude in oceanic basins. Deep Sea Research (1953), 6(3), 217-233.

ID:(12099, 0)


Structure of the Stommel-Arons model
Code

When Stommel and Arons [1], [2] developed their first model of thermohaline circulation, they subdivided the different oceans into zones with defined upwelling (upward arrows) and two sources, one in the Arctic and the other in Antarctica:

Global circulation model in Sv (Sverdrup) ($10^6 m^3/s$) [2].

[1] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanI. Stationary planetary flow patterns on a sphere. Deep Sea Research (1953), 6(2), 140-154.

[2] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanII. An idealized model of the circulation pattern and amplitude in oceanic basins. Deep Sea Research (1953), 6(3), 217-233.

ID:(12098, 0)


True thermohaline circulation
Flux

Measurements have shown that the thermohaline circulation is an integrated system that spans the entire globe. It has at least two points that can be considered as sources, and its path extends across all the oceans.

ID:(12097, 0)


Study of the possible collapse of the deep flow
Matrix

Through multiple simulations, the effects of polar ice melting on the suppression of sinking and its impact on deep circulation are studied. There are indications that circulation has started to decline; however, the collapse of deep circulation does not necessarily imply the same will happen to surface circulation, which is driven by winds. What could happen is a shift in the surface circulation, resulting in a reduction of the Gulf Stream's contribution of warm waters to northern Europe.

The diagram below shows variations in flow in units of Sv (Sverdrup), which is equivalent to $10^6,m^3/s$:

Assuming a sinking rate of approximately 20 Sv, it is concluded that in some simulations, deep circulation comes to a halt. These variations are associated with different future scenarios of human activity and considerations for aspects with less certainty about their occurrence. More detailed information can be found in the reports of the Intergovernmental Panel on Climate Change (IPCC).

ID:(13430, 0)


Deep circulation flows
Description

There are several points where flows from the ocean surface to greater depths occur, inducing deep circulation. This circulation is subject to the Coriolis force, resulting in deviations and some flows towards the surface (upwelling) that are associated with surface currents. The classic model for these currents is that of Stommel and Arons, which, although simple, explains the different depth flows observed. [1] Ocean Circulation Theory, Joseph Pedlosky, Springer 1998 (7.3 Stommel-Arons Theory: Abyssal Flow on the Sphere)

Variables
Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
$\omega$
omega
Angular velocity of the planet
rad/s
$H$
H
Average flow height
m
$U_x$
U_x
Average upwelling flow by latitude
m^3/s
$\Delta t_y$
Dt_y
Characteristic time interval movement in $y$
s
$\Delta t_z$
Dt_z
Characteristic time interval movement in $z$
s
$a_{c,z}$
a_cz
Coriolis acceleration in z direction
m/s^2
$a_{c,x}$
a_cx
Coriolis acceleration on the surface, in the x direction
m/s^2
$a_{c,y}$
a_cy
Coriolis acceleration on the surface, in the y direction
m/s^2
$\beta$
beta
Coriolis Beta Factor
rad/s m
$f$
f
Coriolis factor
rad/s
$x_e$
x_e
Distance east edge and Greenwich meridian
m
$y_n$
y_n
Distance equator north edge
m
$y_s$
y_s
Distance equator south edge
m
$x_w$
x_w
Distance west edge and Greenwich meridian
m
$S_0$
S_0
Inflow
m^3/s
$\varphi$
phi
Latitude
rad
$x$
x
Longitude position
m
$T_w$
T_w
Loss flow
m^3/s
$T_i$
T_i
Main flow
m^3/s
$v_x$
v_x
Parallel speed
m/s
$R$
R
Planet radio
m
$y$
y
Position in latitude
m
$f_0$
f_0
Reference Coriolis factor
rad/s
$v_y$
v_y
Speed in meridian
m/s
$\Delta y$
Dy
Stommel and Arons model case length
m
$\Delta x$
Dx
Stommel and Arons model case width
m
$v_z$
v_z
Upwelling speed
m/s
$v_{zx}$
v_zx
Velocidad de surgencia por meridiano
m/s

Calculations

First, select the equation:   to ,  then, select the variable:   to 
Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

 Variable   Given   Calculate   Target :   Equation   To be used


Equations

As the coriolis acceleration in x direction ($a_{c,x}$) is composed of the angular velocity of the planet ($\omega$), the latitude ($\varphi$), the speed y of the object ($v_y$), and the speed z of the object ($v_z$):

$ a_{c,x} = 2 \omega ( v_y \sin \varphi - v_z \cos \varphi )$



and the definition of the coriolis factor ($f$) is:

$ f = 2 \omega \sin \varphi $



in addition to the constraint of movement on the surface where:

$v_z = 0$



this leads to the coriolis acceleration in x direction ($a_{c,x}$) being:

$ a_{c,x} = f v_y $

(ID 11698)

Since the coriolis acceleration in y direction ($a_{c,y}$) is composed of the angular velocity of the planet ($\omega$), the speed x of the object ($v_x$), and the latitude ($\varphi$):

$ a_{c,y} = -2 \omega v_x \sin \varphi$



and the definition of the coriolis factor ($f$) is:

$ f = 2 \omega \sin \varphi $



in addition to the constraint of movement on the surface where:

$v_z = 0$



this leads to the coriolis acceleration in y direction ($a_{c,y}$) being:

$ a_{c,y} = - f v_x $

(ID 11699)

If we introduce typical timescales for each dimension, we can estimate the Coriolis accelerations as velocities divided by their typical timescales, that is:

$v_i =a_i \Delta t_i$



with i=x,y,z. For the z component, according to coriolis acceleration in z direction $m/s^2$, coriolis Beta Factor $rad/s m$, parallel speed $m/s$ and planet radio $m$, we have:

$ a_{cz} = R \beta v_x $



Thus, we have:

$v_z=\beta R v_x\Delta t_z$



On the other hand, with the equation for the x component of the Coriolis acceleration, which is given by coriolis acceleration on the surface, in the y direction $m/s^2$, coriolis factor $rad/s$ and parallel speed $m/s$, we have, if we neglect the sign:

$v_x=\displaystyle\frac{v_y}{f\Delta t_y}$



By replacing v_x in this previous equation, we obtain with coriolis acceleration on the surface, in the y direction $m/s^2$, coriolis factor $rad/s$ and parallel speed $m/s$:

$ v_z =\displaystyle\frac{ \beta }{ f }\displaystyle\frac{ \Delta t_z }{ \Delta t_y } R v_y $

(ID 12089)

When there is movement in the x-direction (east-west), it results in the coriolis acceleration in z direction ($a_{c,z}$) with the speed x of the object ($v_x$), the angular velocity of the planet ($\omega$), and the latitude ($\varphi$):

$ a_{c,z} = 2 \omega v_x \cos \varphi$



This is complemented by the coriolis acceleration on the surface, in the x direction ($a_{c,x}$) (east-west), with the coriolis factor ($f$) and the speed y of the object ($v_y$):

$ a_{c,x} = f v_y $



and the coriolis acceleration on the surface, in the y direction ($a_{c,y}$) (north-south) with the coriolis factor ($f$) and the speed x of the object ($v_x$), which is:

$ a_{c,y} = - f v_x $



Where the coriolis factor ($f$) is defined as:

$ f = 2 \omega \sin \varphi $



Therefore, we can introduce the coriolis Beta Factor ($\beta$), defined as:

$ \beta =\displaystyle\frac{ 2 \omega \cos \varphi }{ R }$



With this, we obtain:

$ a_{cz} = R \beta v_x $

(ID 12104)

In analogy to the coriolis factor ($f$) defined with the latitude ($\varphi$) and the angular velocity of the planet ($\omega$) as:

$ f = 2 \omega \sin \varphi $



the factor varies in the arc $R\theta$, with the planet radio ($R$) and the latitude ($\varphi$) as the latitude, according to:

$\displaystyle\frac{\partial f}{\partial (R\varphi) }=\displaystyle\frac{ 2\omega\cos\varphi }{R}$



thus the coriolis Beta Factor ($\beta$) can be defined as:

$ \beta =\displaystyle\frac{ 2 \omega \cos \varphi }{ R }$

(ID 12105)

Examples


(ID 15584)

The deeper circulation is known as thermohaline circulation (THC) because its movement is associated with variations in temperature (thermo) and salinity (haline). To understand how this occurs, we must first describe the structure of the system.

In a simplified form, the ocean can be modeled as a three-layer system:

- An upper layer where water movement is generated by wind-driven currents.
- An intermediate layer where movement is driven by density differences in the ocean, originating from variations in temperature and salinity (thermohaline).
- A deep layer that can be assumed to be at rest.

The increase in density towards the poles, where the water is colder, causes the water to literally sink, creating subduction beneath the surface layer. The following diagram summarizes the described process:

(ID 12095)

If we observe the globe, the thermohaline circulation is generated near one of the poles (north or south) through water that, due to higher salinity and lower temperature, begins to sink. Its flow is directed towards the equator, creating an upwelling where water partially rises and flows towards the pole to replenish the descending water.

Representation of the North Atlantic in the Stommel and Arons model [1], [2]

[1] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanI. Stationary planetary flow patterns on a sphere. Deep Sea Research (1953), 6(2), 140-154.

[2] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanII. An idealized model of the circulation pattern and amplitude in oceanic basins. Deep Sea Research (1953), 6(3), 217-233.

(ID 12096)

The Stommel and Arons model [1], [2] considers the ocean as a two-dimensional box with coordinates on the x and y axes. Specifically:

- Coordinates on the x-axis: $x_w$ (west) and $x_e$ (east).
- Coordinates on the y-axis: $y_s$ (south) and $y_n$ (north).

These coordinates are represented in the following graph:

Atlantic box model [1], [2].

[1] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanI. Stationary planetary flow patterns on a sphere. Deep Sea Research (1953), 6(2), 140-154.

[2] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanII. An idealized model of the circulation pattern and amplitude in oceanic basins. Deep Sea Research (1953), 6(3), 217-233.

(ID 12082)

Each stage is associated with a characteristic time:

- Travel time with the main flow $\Delta t_y$
- Deflection time with the loss flow $\Delta t_x$
- Upwelling time $\Delta t_z$

(ID 13426)

Each characteristic time is associated with velocities and accelerations along the traveled path:

- With the main flow $v_y, a_y$.
- With the loss flow $v_x, a_x$.
- With the upwelling $v_z, a_z$.

Generally, the initial velocity ($v_y$) triggers, through the Coriolis force, the accelerations that lead to loss and upwelling.

(ID 13427)

The loss flow is not uniform and is distributed along the latitude, so it is modeled based on its distance from the northernmost position. Thus, it is zero at northern latitudes and maximum at the southern edge of the rectangle where the circulation is modeled:

(ID 13428)

Since the loss flow is not uniform, neither is the upwelling. Within the same model, it is assumed that the upwelling is maximum at the eastern edge of the rectangle where the circulation is modeled. Similar to the loss, a linear relationship is assumed:

(ID 13429)

There are four flows to consider in modeling deep flow:

The main flow $F_w$, which moves along the seafloor.
The loss flow $F_i$, which is the fraction deviated due to the Coriolis force.
The upwelling flow $U_x$, which corresponds to the fraction of loss flow reaching the surface.
The sinking flow $S_0, originating from surface currents, including losses that sink again.

(ID 13425)

The so-called Coriolis force plays an essential role in the dynamics of water in the poles, influencing how water masses descend due to variations in temperature and salinity.



When analyzing the Atlantic Ocean, one can observe a movement of water from the pole towards the equator, which deflects towards the west. This phenomenon is caused by the lag relative to the planet\'s rotation, as it transitions from a zone of slower speed along the latitude to one of higher speed. This behavior can be modeled using the Coriolis equation for the x-direction, given by coriolis acceleration on the surface, in the x direction $m/s^2$, coriolis factor $rad/s$ and speed in meridian $m/s$:

$ a_{c,x} = f v_y $



In this equation, the Coriolis factor f is positive in the Northern Hemisphere and negative in the Southern Hemisphere, generating a tendency for the current to \'approach\' the American continent.

The geographical contour of the continent allows for movement in the x-direction (longitude), resulting in an acceleration in the y-direction (latitude), which can be calculated using coriolis acceleration on the surface, in the y direction $m/s^2$, coriolis factor $rad/s$ and parallel speed $m/s$:

$ a_{c,y} = - f v_x $



This calculation reveals that near the equator, displacements occur that take water away from the main current, moving it northward. If we examine the acceleration in the z-direction (depth) and take into account that \beta also changes sign with the hemisphere, the result is positive. In other words, an upwelling is observed, which can be estimated using coriolis acceleration in z direction $m/s^2$, coriolis Beta Factor $rad/s m$, parallel speed $m/s$ and planet radio $m$:

$ a_{cz} = R \beta v_x $

.

(ID 12122)

At the end, Stommel and Arons [1], [2] solve the model, indicating the main deep flows that exist throughout the globe:

[1] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanI. Stationary planetary flow patterns on a sphere. Deep Sea Research (1953), 6(2), 140-154.

[2] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanII. An idealized model of the circulation pattern and amplitude in oceanic basins. Deep Sea Research (1953), 6(3), 217-233.

(ID 12099)

When Stommel and Arons [1], [2] developed their first model of thermohaline circulation, they subdivided the different oceans into zones with defined upwelling (upward arrows) and two sources, one in the Arctic and the other in Antarctica:

Global circulation model in Sv (Sverdrup) ($10^6 m^3/s$) [2].

[1] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanI. Stationary planetary flow patterns on a sphere. Deep Sea Research (1953), 6(2), 140-154.

[2] Stommel, H., & Arons, A. B. (1960). On the abyssal circulation of the world oceanII. An idealized model of the circulation pattern and amplitude in oceanic basins. Deep Sea Research (1953), 6(3), 217-233.

(ID 12098)

Measurements have shown that the thermohaline circulation is an integrated system that spans the entire globe. It has at least two points that can be considered as sources, and its path extends across all the oceans.

(ID 12097)

Through multiple simulations, the effects of polar ice melting on the suppression of sinking and its impact on deep circulation are studied. There are indications that circulation has started to decline; however, the collapse of deep circulation does not necessarily imply the same will happen to surface circulation, which is driven by winds. What could happen is a shift in the surface circulation, resulting in a reduction of the Gulf Stream's contribution of warm waters to northern Europe.

The diagram below shows variations in flow in units of Sv (Sverdrup), which is equivalent to $10^6,m^3/s$:

Assuming a sinking rate of approximately 20 Sv, it is concluded that in some simulations, deep circulation comes to a halt. These variations are associated with different future scenarios of human activity and considerations for aspects with less certainty about their occurrence. More detailed information can be found in the reports of the Intergovernmental Panel on Climate Change (IPCC).

(ID 13430)


(ID 15585)

ID:(1623, 0)